for all \(p\in M\)  and \(z_1,z_2\in \partial M\) .
These functions give the lengths of the broken geodesics, that is, the
union of the shortest geodesics connecting \(z_1\)  to \(p\)  and the
shortest geodesics connecting \(p\)  to \(z_2\) .
Also, the gradients of \(D^+_p(z_1,z_2)\)  with respect to \(z_1\)  and \(z_2\)  give the velocity
vectors of these geodesics.
The inverse problem of determining the
manifold \((M,g)\)  from the broken geodesic data, consisting of
the initial and the final points and directions, and the total length,
of the broken geodesics, has been considered in [1]}. The authors show that broken geodesic data determine the boundary distance data of any compact smooth manifold of dimension three and higher. Finally they use
the results of [2]}, [3]} to prove that the broken geodesic data determine the Riemannian manifold up to an isometry.
A different variant of broken geodesic data was recently considered in [4]}.
